Problem: $ -1.\overline{7} \div 0.\overline{1} = {?} $
First convert the repeating decimals to fractions. $\begin{align*} 10x &= -17.7778...\\ x &= -1.7778...\end{align*} $ $\begin{align*} 9x &= -16 \\ x &= -\dfrac{16}{9}\end{align*} $ $\begin{align*} 10y &= 1.1111...\\ y &= 0.1111...\end{align*} $ $\begin{align*} 9y &= 1 \\ y &= \dfrac{1}{9}\end{align*} $ So, the problem becomes: $ -\dfrac{16}{9} \div \dfrac{1}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{16}{9} \times \dfrac{9}{1} = {?} $ $ \phantom{-\dfrac{16}{9} \times \dfrac{1}{9}} = \dfrac{-16 \times 9}{9 \times 1} $ $ \phantom{-\dfrac{16}{9} \times \dfrac{1}{9}} = \dfrac{-16 \times \cancel{9}} {\cancel{9} \times 1} $ $ \phantom{-\dfrac{16}{9} \times \dfrac{1}{9}} = -\dfrac{16}{1} $